# On finding all positive integers $a,b$ such that $b\pm a$ and $ab$ are palindromic

## TL;DR

This paper characterizes all positive integer solutions where the sums or differences of the pair and their product are palindromes, providing explicit formulas and finite solutions.

## Contribution

It completely classifies solutions for the palindromic sum/difference and product conditions, extending previous partial results.

## Key findings

- Solutions for $a+b$ and $ab$ are explicitly listed.
- Solutions for $b-a$ and $ab$ are given in parametric form.
- Finite and infinite families of solutions are identified.

## Abstract

It is proven that the only integer solutions $(a,b)$ such that $a+b$ and $ab$ are palindromic are $(2,5\cdot 10^k-3)$, $(3,24)$ and $(9,9)$, and in a similar fashion, $b-a$ and $ab$ are only palindromic at $(a,b)=(3,147\cdot 10^{4(k+1)}+5247\sum_{i=0}^k10^{4i})$, $(3,161\,247\cdot 10^{4k+7}+5247\sum_{i=0}^k10^{4i+3}+387)$, $(3,147)$ and $(3,161\,247\,387)$ for $k=0,1,2,\cdots$. Note $a\le b$ without loss of generality.